Collective excitations in three-dimensional Dirac systems

We provide the plasmon spectrum and related properties of the three-dimensional (3D) Dirac semimetals ${\mathrm{Na}}₃\mathrm{Bi}$ and ${\mathrm{Cd}}₃{\mathrm{As}}₂$ based on the random-phase approximation. The necessary one-electron eigenvalues and eigenfunctions are obtained from an effective $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonian. Below the energy at which the velocity ${v}_z$ along the ${k}_z$ axis vanishes, the density of states differs drastically from that of a 3D electron gas (3DEG) or graphene. The dispersion relation is anisotropic for wave vectors parallel ($q$) and perpendicular (${q}_z$) to the $(x,y)$ plane and is markedly different than that of graphene or a 3DEG. The same holds for the energy-loss function. Both depend sensitively on the position of the Fermi energy ${E}_F$ relative to the region of the Berry curvature of the bands. For ${E}_F$ below the energy at which ${v}_z$ vanishes, the range of the relevant wave vectors $q$ and ${q}_z$ shrinks, for ${q}_z$ by about one order of magnitude.